On the Equivalence of the Melnikov Functions Method and the Averaging Method

We prove the equivalence between the Melnikov functions method and the averaging method as tools for finding limit cycles of analytic planar differential systems which are perturbations of a period annulus. We consider any possible change of variables to transform the planar system into a scalar periodic equation which perturbs a continuum of constant solutions. We prove that the Poincar, return map of the planar system and the Poincar, translation map of the scalar equation coincide. For distinct specific changes of variables this was stated before in 2004 by Buic-Llibre and proved in 2015 by Han-Romanovski-Zhang.